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Boundary layer flow past a stretching/shrinking
surface beneath an external uniform shear flow
with a convective surface boundary condition in
a nanofluid
Nor Azizah Yacob
1
, Anuar Ishak
2
, Ioan Pop
3*
and Kuppalapalle Vajravelu
4
Abstract
The problem of a steady boundary layer shear flow over a stretching/shrinking sheet in a nanofluid is studied
numerically. The governing partial differential equations are transformed into ordinary differential equations using a
similarity transformation, before being solved numerically by a Runge-Kutta-Fehlberg method wi th shooting
technique. Two types of nanofluids, namely, Cu-water and Ag-water are used. The effects of nanoparticle volume
fraction, the type of nanoparticles, the convective parameter, and the thermal conductivity on the heat transfer
characteristics are discussed. It is found that the heat transfer rate at the surface increases with increasing
nanoparticle volume fraction while it decreases with the convective parameter. Moreover, the heat transfer rate at
the surface of Cu-water nanofluid is higher than that at the surface of Ag-water nanofluid even though the
thermal conductivity of Ag is higher than that of Cu.
Introduction
Blasius [1] was the first who studied the steady bound-
ary layer flow over a fixed flat plate with uniform free
stream. Howarth [2] solved the Blasius problem numeri-
cally. Since then, many researchers have investigated the
similar problem with various physical aspects [3-6].
In contrast to the Blasius problem, Sakiadis [7] intro-
duced the boundary layer flow induced by a moving

plate in a quiescent ambient fluid. Tsou et al. [8] studied
the flow and temperature fields in the boundary layer on
a continuous moving surface, both analytically and
experimentally and ve rified the results obtained in [7].
Crane [9] extended this c oncept to a stretching plate in
aquiescentfluidwithastretching velocity that varies
with the distance from a fixed point and pre sented an
exact analytic solution. Different from the above studies,
Miklavčič andWang[10]examinedtheflowduetoa
shrinking sheet where the velocity moves toward a fixed
point. Fang [11] studied the boundary layer flow over a
shrinking sheet with a power-l aw velocity, and obtained
exact solutions for some values of the parameters.
It is well known that Choi [12] was the first to intro-
duce the term “nanofl uid” that represents the fluid in
which nano-scale particles are suspended in the base
fluid with low thermal conductivity such as water, ethy-
lene glycol, oils, etc. [13]. In recent years, the concept of
nanofluid has been proposed as a route for surpassing
the performance of heat transfer rate in liquids currently
available. The materials with sizes of nanometers possess
unique physical and chemical properties [14]. They can
flow smoothly through microchannels without clogging
them because they are small enough to behave similar
to liquid molecules [15]. This fact has attracted many
researchers such as [16-27] to investigate the heat trans-
fer characteristics in nanofluids, and they found that in
the presence of the nanoparticles in the fluids, the effec-
tive thermal conductivity of the fluid increases appreci-
ably and consequently enhances the heat transfer

characteristics. An excelle nt collection of articles on this
topic can be found in [28-33], and in the book by Das
et al. [14].
* Correspondence:
3
Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania
Full list of author information is available at the end of the article
Yacob et al. Nanoscale Research Letters 2011, 6:314
/>© 2011 Yacob et al; licensee Springer. This is an Open Access article d istribute d under the terms of the Creative Commons Attribution
License ( which perm its unrestricted use, distribution , and reproduction in any medium,
provided the original work is properly cited.
It is worth mentioning that while modeling the
boundary layer flow and heat transfer of stretching/
shrinking surfaces, the boundary conditions that are
usually applied are either a specified surface temperature
or a specified surface heat flux. However, there are
boundary layer flow and heat transfer problems in
which the surface heat transfer depends on the surface
temperature. Perhaps the simplest case of this is whe n
there is a linear relatio n between the surface heat trans-
fer and surface temperature. This situation arises in con-
jugate heat transfer problems (see, for example, [34]),
and when there is Newtonian heating of the convective
fluid from the surface; the latter c ase was discussed in
detail by Merkin [35]. The situation with Newtonian
heating arises in what is usually termed as conjugate
convective flow, where the heat is supplied to the con-
vective fluid through a bounding surface with a finite
heat capacity. This results in the heat transfer rate
through t he surface being proportional to the local dif-

ference in the temperature with the ambient conditions.
This configuration of Newtonian heating occurs in
many important engineering devices, for example, in
heat exchangers, where the conduction in a solid tube
wall is greatly influenced by the convection in the fluid
flowing over it. On the other hand, most recently, heat
transfer problems for boundary layer flow concerning
with a convective boundary condition were investigated
by Aziz [36], Makinde and Aziz [37], Ishak [38], and
Magyari [39] for the Blasius flow. Similar analysis was
applied to the Blasius and Sakiadis flows with radiation
effects by Bataller [4]. Y ao et al. [40] have very recently
investigated the heat transfer of a viscous fluid flow over
a permeable stretching/shrinking sheet with a convective
boundary condition. Magyari and Weidman [41] investi-
gated the hea t transfer characteristi cs on a semi-infinite
flat plate due to a unifo rm shear flow, both for the pre-
scribed surface temperature and prescribed surface heat
flux. It is worth pointing out that a uniform shear flow
is driven by a viscous outer flow of rotational velocity
whereas the classical Blasius flow is driven over the
plate by an inviscid outer flow of irrotational velocity.
The objective of this study is to extend the study of
Magyari and Weidman [41] to a stretching/shrinking
surface with a convective boundary condition immersed
in a nanofluid, that is, to study the steady boundary
layer shear flow over a stretching/shrinking surface
beneath an external uniform shear flow with a convec-
tive surface boundary condition in a nanofluid. This
problem is relevant to several practical applications in

the field of metallurgy, chemical engineering, etc. A
number of technical processes concerning polymers
involve the cooling of continuous strips or filaments by
drawing them through a quiescent fluid. In these cases,
the properties of the final product depend to a great
extent on the rate of cooling, which is governed by the
structure of the boundary layer near the stretching/
shrinking surface. The governing partial differential
equations are transformed into ordinary differential
equations using a similarity transformation, before being
solved numerically by the Runge-Kutta-Fehlberg method
with shooting technique.
Mathematical formulation
Consider a two-dimensional steady boundary layer shear
flow over a stretching/shrinking shee t in a laminar and
incompressible nanofluid of ambient temperature T
∞
.
The fluid is a water-based nanofluid containing two type
of nanoparticles, either Cu (copper) or Ag (silver). The
nanoparticles are assumed to have a uniform shape and
size. Moreover, it is assumed that both the fluid phase
and nanoparticles are in t hermal equilibrium state.
Figure 1 describes the physical model and th e coordi-
nate system, where the x and y axes are measured along
the surface of the sheet and normal to it, respectively.
Following Magyari and Weidman [41], it is assumed
that the velocity of the moving stretching/shrinking
sheet is u
w

(x)=U
w
(x/L )
1/3
and the velocity outside the
boundary layer (potential flow) is u
e
(y)=by,whereb is
the constant strain rate. We also assume that the bot-
tom surface of the stretching/shrinking surface is heated
by convection from a base (water) fluid a t temperature
T
f
, which provides a heat transfer coefficient h
f
(see
[36]). Under the boundary layer approximations, the
basic equations are (see [17,42]),
∂
u
∂x
+
∂
v
∂
y
=
0
(1)
u

∂u
∂x
+ v
∂u
∂
y
=
μ
nf
ρ
nf
∂
2
u
∂
y
2
(2)
u
∂T
∂x
+ v
∂T
∂
y
= α
nf
∂
2
T

∂
y
2
(3)
Further, we assume that the sh eet surface temperature
is maintained by convective heat transfer at a constant
temperature T
w
(see [36]). Thus, the boundary condi-
tions of Equations 1-3 are
v =0, u = u
w
(x)=U
w

x
L

1/3
, k
f

∂T
∂y

= h
f
(T
f
− T

∞
)aty =
0
u = u
e
(
y
)
= βy, T = T
∞
as y →∞
(4)
where L is the c haracte ristic length of the stretching/
shrinking surface. The properties of nanofluids are
defined as follows (see [20]):
Yacob et al. Nanoscale Research Letters 2011, 6:314
/>Page 2 of 7
α
nf
=
k
nf
(ρC
p
)
nf
, ρ
nf
=(1− ϕ)ρ
f

+ ϕρ
s
, μ
nf
=
μ
f
(1 − ϕ)
2.5
(ρC
p
)
nf
=(1− ϕ)(ρC
p
)
f
+ ϕ(ρC
p
)
s
,
k
nf
k
f
=
(k
s
+2k

f
) − 2ϕ(k
f
− k
s
)
(
k
s
+2k
f
)
+ ϕ
(
k
f
− k
s
)
(5)
Following Magyari and Weidman [41] and Aziz [36],
we look for a similarit y solution of Equations 1-3 of the
form:
ψ = ν
f

x
L

1/3

f (η), θ(η)=
T − T
∞
T
f
− T
∞
, η =

x
L

−1/3
y
L
(6)
where ν
f
is the kinematic viscosity of the base (water)
fluid, and ψ is the stream function, which is defined as
u= ∂ψ/∂y and v = –∂ψ/∂x, which automatically satisfies
Equation 1. A simple analysis shows that L =(ν
f
/b)
1/2
.
Substituting (6) i nto Equations 2 and 3, we obtain the
following ordinary differential equations:
3
(

1 − ϕ
)
2.5
(
1 − ϕ + ϕρ
s
/ρ
f
)
f

+2ff

− f
2
=
0
(7)
3
Pr
k
nf
/k
f

1 − ϕ + ϕ(ρC
p
)
s
/(ρC

p
)
f

θ

+2fθ

=
0
(8)
subject to the boundary conditions
f (0) = 0, f

(0) = λ, θ

(0) = −γ

1 − θ(0)

f

(
η
)
= η, θ
(
η
)
=0 asη →∞

(9)
where primes denote differentiation with respect to h,
and l = U
w
/(bν
f
)
1/2
is the stretching/shrinking para-
meter, and g is given by
γ =
h
f
L
k
f

x
L

1/
3
(10)
For the thermal equation (8) to have a similarity solu-
tion, the quantity g must be a constant and not a func-
tion of x as in Equation 10. This condition can be met if
h
f
is proportional to (x/L)
-1/3

. We, therefore, assume
h
f
= c

L
x

1/
3
(11)
where c is a constant. Thus, we have
γ = cL
/
k
f
(12)
with g defined by Equation 12, the solutions of Equa-
tions 7-9 yield the similarity solutions. However, with g
defined by Equation 10, the generated solutions are
local similarity solutions. We notice that the solution of
Equations 7 and 8 approaches the solution for the con-
stant surface temperature as g ® ∞. This can be seen
from the boundary conditions (9), which gives θ(0) = 1
as g ® ∞. Further, it is worth mentioning that Equa-
tions 7 and 8 redu ce to those of Magyari and Weidman
[41] when  = 0 (regular fluid) and l = 0 (fixed surface).
The quantities of interest are the skin friction coeffi-
cient C
f

and the local Nusselt number Nu
x
, which repre-
sents the heat transfer rate at the surface, and they can
be shown to be given in dimensionless form as

L
2/3
U
w
x
1/3
ν
f

2
C
f
=
1
(
1 − ϕ
)
2.5
f

(0),

L
x


2/3
Nu
x
= −
k
nf
k
f
θ

(0
)
(13)
Results and discussion
The nonlinear ordinary differential equations (7) and (8)
subject to the boundary conditions (9) were solved
numerically by the Runge-Kutta-Fehlberg method with
w
T

I
ncoming
shear flow
()
e
uuy
TT
f

Nanofluid

Hot fluid
,,
fff
Thk
y
x
()
w
uux

O

nf s f
(, ),kkkT
f

Figure 1 Physical model and the coordinate system.
Yacob et al. Nanoscale Research Letters 2011, 6:314
/>Page 3 of 7
shooting technique. We consider two different types of
nanoparticles, namely, Cu and Ag with w ater as the
base fluid. Table 1 shows the thermophysical properties
of water and the elements Cu and Ag. The Prandtl
number of the bas e fluid (water) is kept constant at 6.2.
It is worth mentioning that this study reduces to those
of a viscous or regular fluid when  = 0. Figure 2 shows

the variation of the skin friction coefficient ( 1/(1-)
2.5
)
f”(0) with l o f Ag-water nanofluid when g = 0.5 for dif-
ferent nanoparticle volume fraction , while the respec-
tive local Nusselt number -(k
nf
/k
f
) θ’ (0) is displayed in
Figure 3. It can be seen that for a particular value of l,
the s kin friction coefficient and the local Nusselt num-
ber increase with increasing . Dual solutions are found
to exist when l < 0 (shrinking case) as displayed in
Figures 2 and 3. Moreover, the solution can be obtained
up to a critical value of l (say l
c
), and |l
c
| decreases
with increasing . The similar pattern is observed for
Cu-water nanofluid, which is not presented here, for the
sake of brevity. It is observed that, the solution is unique
for l ≥ 0, dual solutions exist for l
c
< l < 0, and no
solution for l <l
c
.Thevaluesofl
c

forAg-waternano-
fluid and Cu-water nanofluid for different values of 
are presented in Table 2. It is seen that for  =0.1and
 =0.2,thevalueof|l
c
| for Cu-water nanofluid is
greater than those of Ag-water nanofluid. The tempera-
ture profiles of Ag-water a nd Cu-water nanofluids for
different values of  when g = 0.5 and l = -0.53 are pre-
sented in Figures 4 and 5, respectively. These profiles
show that, there exist tw o different profiles satisfying
the far field boundary condition (9) asymptotically, thus
supporting the dual nature of the solutions presented in
Figures 2 and 3. Both Figures 4 and 5 show that the
boundary layer thickness is higher for the second solu-
tion compared to the first solution, which in turn pro-
duces higher surface temperature θ(0) for the former.
Figure 6 displays the variation of the skin friction
coefficient (1/(1-)
2.5
)f” (0) with l when g =0.5for
water, Cu-water and Ag-water nanofluids, while the
respective local Nusselt number -(k
nf
/k
f
)θ’(0) is shown in
Figure 7. In general, for a particular value of l, the ski n
fri ction coefficient of Cu-water nan ofluid is higher than
that of Ag-water nanofluid and that of water for the

upper branch solutions, while the skin friction coeffi-
cient of Ag-water nanofluid is higher than that of Cu-
water nanofluid and that of water for the lower branch
solutions. Further, Figure 7 shows that Cu-water nano-
fluid has the highest local Nusselt number compared
with Ag-water nanofluid and water for the upper branch
solutions. From this observation, the heat transfer rate
at the surface of Cu-water nanofluid is higher than that
of Ag-water nanofluid even though Ag has higher ther-
mal conductivity than the thermal conductivity of Cu as
Table 1 Thermophysical properties of water and the
elements Cu and Ag
Physical Properties Fluid Phase (Water) Cu Ag
C
p
(J/kgK) 4179 385 235
r (KG/m
3
) 997.1 8933 10500
k (W/mK) 0.613 400 429
a ×10
7
(m
2
/s) 1.47 1163.1 1738.6
Figure 2 Variation of the skin fri ction coefficient with l for
different values of  when g = 0.5 for Ag-water nanofluid.
-(k
nf
k

f
)Tc(0)
Figure 3 Variation of the local Nusselt number with l for
different values of  when g = 0.5 for Ag-water nanofluid.
Table 2 Values of l
c
for Cu-water and Ag-water
nanofluids
 l
c
Cu Ag
0 -0.62228 -0.62228
0.1 -0.55512 -0.53870
0.2 -0.53929 -0.51800
Yacob et al. Nanoscale Research Letters 2011, 6:314
/>Page 4 of 7
presented in Table 1. However, the difference in heat
transfer rate at the surface is small. On the other hand,
Ag-water nanofluid has the highest local Nusselt num-
ber compared with Cu-water nanofluid and water for
the lower branch solutions. The corresponding tempera-
ture profiles that support the results obtained in Figure
7 when l = -0.53 is shown in Figure 8.
It is observed from Figures 2, 3, 6, and 7 that the skin
friction coefficient and the local Nusselt number are
more influenced by the nanoparticle volume fraction
than the types of nanoparticles. This observation is in
agreem ent with those obtained by Oztop and Abu-Nada
[20] and Abu-Nada and Oztop [43]. In addition, water
has the lowest skin friction coefficient and local Nusselt

number compared with Cu-water and Ag-water nano-
fluids. The range of l for which the solution exists is
wider for water compared with the others.
The temperature profiles of Ag-water nanofluid for dif-
ferent values of convective parameter g when  =0.2is
presented in Figure 9. It is observed that the surface tem-
perature increases with an increase in g for both solution
branches, and in consequence, decreases the loc al Nus-
selt number. It can be seen that from the convective
boundary conditions (9), the value of θ(0) approaches 1,
as g ® ∞. Further, the convective parameter g as well as
the Prandtl number Pr has no influence on the flow field,
which is clear from Equations 7-9. Finally, it is worth
mentioning that all the velocity and temperature pro files
Figure 4 Temperature profiles for Cu-water nanofluid when g =
0.5 and l = -0.53 for different values of .
Figure 5 Temperature profiles for Ag-water nanofluid when g =
0.5 and l = -0.53 for different values of .
Figure 6 Variation of the skin friction coefficient with l when g
= 0.5 and  = 0.1 for different nanofluids and water.
-(k
nf
k
f
)Tc(0)
Figure 7 Variation of the local Nusselt number with l when
g = 0.5 and  = 0.1 for different nanofluids and water.
Yacob et al. Nanoscale Research Letters 2011, 6:314
/>Page 5 of 7
presented in Figures 4, 5, 7, 8, and 9 satisfy the far-field

boundary conditions (9) asymptotically, thus supporting
the validity of the numerical results obtained.
Conclusions
The problem of a steady boundary layer shear flow over a
stretching/shrinking sheet in a nanofluid was studied
numerically. The governing partial differential equations
were transformed into ordinary different ial equations by
a similarity transformation, before being solved numeri-
cally using the Runge-Kutta-Fehlberg method with shoot-
ing technique. We considered two types of nanofluids,
namely, Cu-water and Ag-water. It was found that the
heat transfer rate at the surface increases with increasing
nanoparticle volume fraction while it decreases with the
convective parameter. The variations of the skin friction
coefficient and the heat transfer rate at the surface are
more influenced by the nanoparticle volume fraction
than the types of the nanofluids. Moreover, the heat
transfer rate at the surface of Cu-water nanofluid is
higher than that of the Ag-w ater nanofluid even though
Ag has higher thermal conductivity than that of Cu.
Abbreviations
List of symbols: c: Constant; C
f
: Skin friction coefficient; C
p
: Specific heat at
constant pressure; f: Dimensionless stream function; h
f
: Heat transfer
coefficient; k: Thermal conductivity; L: Reference length; Nu

x
: Local Nusselt
number; Pr: Prandtl number; q
w
: Surface heat flux; T: Fluid temperature; T
f
:
Temperature of the hot fluid; T
w
: Surface temperature; T
∞
: Ambient
temperature; u, v: Velocity components along the x and y-directions,
respectively; u
e
(y): Free stream velocity; u
w
(x): Stretching/shrinking velocity;
U
w
: Reference stretching/shrinking velocity; x, y: Cartesian coordinates along
the surface and normal to it, respectively; Greek symbols: α: Thermal
diffusivity; β: Constant strain rate; γ: Convective parameter; η: Similarity
variable; θ: Dimensionless temperature; λ: Stretching/shrinking parameter; μ:
Dynamic viscosity; ν: Kinematic viscosity; ρ: Fluid density; : Nanoparticle
volume fraction; ψ: Stream function; τ
w
: Wall shear stress; Subscripts; f: Fluid;
nf: Nanofluid; s: Solid.
Acknowledgements

The authors express their sincere thanks to the anonymous reviewers for their
valuable comments and suggestions for the improvement of the article. This
study was supported by research grants from the Ministry of Science,
Technology and Innovation, Malaysia (Project Code: 06-01-02-SF0610) and the
Universiti Kebangsaan Malaysia (Project Code: UKM-GGPM-NBT-080-2010).
Author details
1
Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA
Pahang, 26400 Bandar Tun Razak Jengka, Pahang, Malaysia
2
Centre for
Modelling & Data Analysis, School of Mathematical Sciences, Faculty of
Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi,
Selangor, Malaysia
3
Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP
253, Romania
4
Department of Mathematics, University of Central Florida,
Orlando, FL 32816, USA
Authors’ contributions
NAY and AI performed the numerical analysis and wrote the manuscript. IP
carried out the literature review and co-wrote the manuscript. KV helped to
draft the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 19 November 2010 Accepted: 7 April 2011
Published: 7 April 2011
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doi:10.1186/1556-276X-6-314
Cite this article as: Yacob et al.: Boundary layer flow past a stretching/
shrinking surface beneath an external uniform shear flow with a
convective surface boundary condition in a nanofluid. Nanoscale
Research Letters 2011 6:314.

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